[Zhang
2011] *Counting Carmichael numbers with small seeds*,
Mathematics
of Computation, **80**:273 (2011),
437-442.

**ABSTRACT**. Let *A*_{s}* *be the product of the first *s** *primes, let *P*_{s}* *be the set of primes *p** *for which *p**−*1 divides *A** _{s}* but

The main purpose of this paper is to give
numerical evidence to support the following conjecture which shows that* *|*C** _{s}*| grows rapidly on

(1) log_{2}* *|*C** _{s}*| = 2

with lim _{s→∞}* **ε
*= 0*.* We describe a procedure to compute
exact values of * *|*C** _{s}*|

|*C*_{9}|* *= 8*,
*281*, *366*, *855*,
*879*, *527 with *ε
*= 0*.*36393
*...*

and that

|*C*_{10}|* ** *=
21*, *823*, *464*,
*288*, *660*, *480*,
*291*, *170*, *614*,
*377*, *509*, *316 with *ε *=
0*.*31662
*...*.

The entire calculation for computing* *|*C** _{s}*|

(Note that the counts of the number of Carmichael numbers in either ErdÖs’s
conjecture or **AGP**’s theorem are
functions which grow slowly on *x* For *x** *= 10^{n}* *for *n** *up
to 21 (which is as far as has been computed by Pinch),
there are fewer than *x*^{0}^{.}^{348}^{ }Carmichael numbers up to *x*.)

*Remark* 1.1. Alford took*
L* = 2^{6} ・ 3^{3} ・ 5^{2} ・ 7^{2} ・ 11, determined 155 primes *p* for which *p* − 1 divides* L*, and then established that there are at least 2^{128} − 1 Carmichael numbers made up from them. However, Alford did not
express the number of Carmichael numbers as a function of *L*. Granville mentioned: “It can be shown that if*
L* = *A** _{s}* for some sufficiently large

(2)
*L *ln^{2}* L*

Carmichael numbers in* C** _{s}*.” The estimate (2) seems to be the only estimate for